We attempted to characterize the basins of attraction of the closed loop system under MPC.
In nonlinear systems, the boundaries of basins of attraction may be smooth or they may be fractal [1]}, [2]}, [3]}.
These basin boundaries are often associated with saddle-type invariant objects (e.g. stable manifolds of saddle steady states/fixed points).
We used numerical experiments to examine whether the basin boundaries in our system are smooth or fractal, and visualized the results with 2D projections of the 4D phase space.
Figure REF  shows initial conditions in \(\mathbf {u}\) -space (top row) and \(\mathbf {y}\) -space (bottom row) colored by the steady state to which they are attracted; in particular, we visualize regions very close to the basin boundaries (in the far right plots).
In both the \((u_1,u_2)\)  plane and \((y_1,y_2)\)  plane, we do not observe any evidence of a fractal basin of attraction, suggesting that the basin boundaries are, to the best of our ability to discern, smooth.
We were not able to numerically locate any additional saddle-type, unstable invariant objects that organize these basin boundaries.
<FIGURE>